This question hinges on two things:
1) Who is doing the requesting?
2) What is the idiomatic form of 'requesting'
The answer to 1) is the Greek authorities - they want the British Museum return the Elgin marbles. So we need an answer choice that clearly shows this relationship. Based on this information let's eliminate.A) requesting that
This answer suggests that the British are doing the requesting. Notice that the subject of the main clause is 'British museum.' The verb phrase "continue to rebuff" refers to British Museum. So when you place the gerund 'requesting' it refers back to British Museum, which is clearly wrong. B) requesting them
This answer choice makes a similar error with the use of 'requesting.' Also, British Museum, the one who should return the Elgin marbles, is a singular entity and thus requires 'it' not 'them.'C) its request
Here we are implying that the British museum is doing the requesting
For point 2) we want the construction, "request that..." This is an example of the subjunctive voice, which is a person/thing is desiring something. Typically words in this tense, such as request, require the use of 'that.'
With this knowledge, we can clearly get rid of (E) who request them to.
Notice that (E) also use 'them' when it should be using 'it'.
Therefore the answer is (D) who request that it.
Note that the 'who' clearly modifies Greek authorities (remember the touch rule).
Hope that helped
Magoosh Test Prep